Horizontal Curve Calculator

Enter the radius in any length unit and the angle in decimal degrees to obtain tangent length, arc length, and external distance. Every returned length uses the same unit as the radius, which makes the result easy to apply to feet, metres, or another consistent project unit without a separate conversion step.

Understanding the three calculated curve elements

A simple circular horizontal curve joins two straight tangents with an arc of constant radius. The tangent length is the distance from the point of intersection of the tangents to either tangent point, measured along the straight alignment. The curve length is the distance travelled along the circular arc between those tangent points. The external distance is measured from the tangent intersection toward the midpoint of the curve and shows how far the arc lies outside the intersection-to-center relationship. Together, these values support preliminary alignment layout, stationing checks, right-of-way studies, and independent verification of design worksheets. This calculator assumes that the supplied deflection angle is the central angle of the simple curve, which is also the change in tangent direction for the standard geometry. It does not design transition spirals, compound curves, reverse curves, superelevation, sight distance, or speed criteria. Those require additional inputs and engineering judgment. Keep the radius and every downstream distance in one consistent unit so that the numerical results retain a clear physical meaning.

How the formulas use radius and deflection angle

The calculator converts the deflection angle from degrees to radians before evaluating the geometry. Tangent length is computed as the radius multiplied by the tangent of one-half the deflection angle. Arc length is the radius multiplied by the full angle in radians, which is equivalent to multiplying radius, pi, and the angle in degrees, then dividing by 180. External distance is the radius multiplied by the secant of one-half the angle minus one. These are exact relationships for a simple circular curve when the radius is measured to the curve centerline. The angle must be greater than zero and less than 180 degrees because the limiting values either collapse the curve or make the tangent and external-distance expressions singular. The implementation uses deterministic floating-point arithmetic and reports stable numeric results suitable for calculations and software workflows. For construction documents, round only after considering the precision of the survey control, the selected units, applicable standards, and the tolerances required by the project rather than assuming that every displayed digit is field accuracy.

Using and checking the results in practice

Start with a radius and deflection angle taken from the same alignment definition. If the radius is entered in metres, tangent length, curve length, and external distance are all returned in metres; if it is entered in feet, all three are returned in feet. A useful reasonableness check is that curve length should increase directly with either radius or angle. For angles below 180 degrees, tangent length should be positive, and the external distance should also be positive. Small angles produce a comparatively shallow offset, while angles approaching 180 degrees make tangent length and external distance grow rapidly. When comparing the answer with a plan set, confirm that the plan uses a simple centerline curve and the same definition of deflection angle. Some transport standards describe curves through degree of curve, chord definitions, or station equations, which need conversion before these formulas apply. Use the output as a transparent geometry calculation, then have a qualified professional review any safety-critical alignment, staking plan, or final design decision under the governing road, rail, and surveying requirements.